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Household Members’ Time Allocation to Daily Activities and Decision to Hire Domestic Helpers

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Research Background

Discrete choice models

The allocation of maintenance activities to household members in terms of who and how (jointly or independently) the maintenance activities are conducted (Vovsha, et al. 2004; Srinivasan and associates 2005;2006)

Research Background

Existing studies share an underlying assumption: no external help for household maintenance tasks

In reality, households may get helps from members of the extended family, or hire domestic helpers

External helps may change activity-travel patterns of household members

Consideration of external helps is important for understanding activity-travel behaviors of household members

Hardly any study of this kind is reported in the literature

Research Objectives

Develop a model to analyze time allocation of household members taking into consideration of hiring domestic helpers.

Calibrate the model by empirical data

Apply the model for analyzing the impacts of wage rate changes on time allocation

Notations

is the total time available for member i;

is the time spent for activity j by member i.

is the amount of time that a domestic helper is hired.

is the baseline utility for time invested in activity j by household member i .

is the baseline utility for hiring domestic helpers

is the wage rate of household member i, is the wage rate of domestic helpers.

Utility Functions of Household Members and Domestic Helpers

We define the utility function of a household member as follows:

And the utility function of hiring a domestic helper is:

A Model of Time Allocation

Households are assumed to maximize the household utility:

(1)

Subject to: (a)

(b)

Where and are respectively weights of household members and domestic helpers, representing their relative contributions to household utility.

A Model of Time Allocation

Lagrangian function

where are respectively the Lagrangian multipliers associated with the time constraint for member and that associated with the time constraint of maintenance.

A Model of Time Allocation

The first-order conditions of the Lagrangian function

A Model of Time Allocation

From the first-order conditions, we may derive the following interrelations among time allocation to activities and time of hiring domestic helper:

(2)

(3)

With constraints (a) and (b)

Specification of the baseline utility function

In general the utility function can be expressed as follows:

(4)

Where are error terms, is the systematic component of utility and assume that it is continuously differentiable

We assume that

The utility contribution of time allocation is dependant on socio-economic characteristics of individuals, the systematic component of utility is to capture these effects

Model Estimation

Substituting equation (4) into (2)and into (3), we got the following equation

Model Estimation

Where

Model Estimation

Assume that error term are independently and identically standard normal distributed

we add the following auxiliary random variable:

based on the property of normally distributed variables, Then the density function of the random vector may be derived as follows:

Model Estimation

Where

Model Estimation

Therefore the density function of the random vector can be derived as follows:

Where

Model Estimation

The maximum likelihood method may be applied to estimate the parameters. Introducing the index for household, we can write the likelihood function as follows:

Where

The sequential number theoretic optimization (SNTO) algorithm can be used to solve the likelihood function to find the maximum likelihood estimation of parameters . SNTO searches for the global optimum among points uniformly scattered in the search space (Fang and Wang, 1994).

An Empirical Study

Hong Kong: A special Administrative region of China

Area size: 1100 square kilometers

High population density (more than 6,000 per square kilometers)

In 2004, about 218,430 foreign domestic helpers working in Hong Kong

An Empirical Study

Hong Kong Travel Characteristics Survey conducted in 2002; only households that have married couples and at least one of the couple is employed are selected;

About the sample:

size: 10,381 households;

About 9.9% have live-in maids;

About 15.9% have car available for private use;

About 17.0% have child aged 1 to 5 and 45.4% have child aged 6-17;

About 48.2% of the households live in private apartment or house

About 76.1% of male heads are employed and 49.2% of female heads are employed.

An Empirical Study

Variable definitions

Time duration of the following activities: subsistence, maintenance and recreation

Wage rate: calculated from monthly household income (divided by duration of subsistence activity)

If both household heads are working: Male : female (1.35 :1)

If only one head is working, all household income is allocated to the person

Domestic helper:

Monthly costs of a domestic helper: around 6000 HK dollars

Working time: 10 hour/day,

Wage rate: 20 HKD/hour

An Empirical Study

Socio-Economic variables

household size,

type of housing (‘1’ for private housing and ‘0’ for public housing),

the presence of child aged 5 or younger and of child aged 6 to 17,

age of household head (1: 20 years old or younger, 2: 21 – 40, 3: 41 – 60, 4: older than 60)

car ownership.

Assume the baseline utility is defined as follows:

Parameter Estimation

Estimation of parameters for subsistence activity

Parameter Estimation

Estimation of parameters for maintenance activity

Parameter Estimation

Estimation of parameters for recreation activity

Number of observations=10381

Model Prediction

The model can be used to predict the time use of household members and time of hiring domestic helpers for household.

Subject to:

Model Prediction

Based on the parameters estimation, we can forecast the time allocation to activities for household members and the time of hiring domestic helper